1

Behaviour and Design of Cold-formed Steel Beams Subject to Lateral-Torsional Buckling at Elevated

Temperatures

Abstract Cold-formed steel beams are increasingly used as floor joists and bearers in buildings and

often their behaviour and moment capacities are influenced by lateral-torsional buckling.

With increasing usage of cold-formed steel beams their fire safety design has become an

important issue. Fire design rules are commonly based on past research on hot-rolled steel

beams. Hence a detailed parametric study was undertaken using validated finite element

models to investigate the lateral-torsional buckling behaviour of simply supported cold-

formed steel lipped channel beams subjected to uniform bending at uniform elevated

temperatures. The moment capacity results were compared with the predictions from the

available ambient temperature and fire design rules and suitable recommendations were

made. European fire design rules were found to be over-conservative while the ambient

temperature design rules could not be used based on single buckling curve. Hence a new

design method was proposed that includes the important non-linear stress-strain

characteristics observed for cold-formed steels at elevated temperatures. Comparison with

numerical moment capacities demonstrated the accuracy of the new design method. This

paper presents the details of the parametric study, comparisons with current design rules and

the new design rules proposed in this research for lateral-torsional buckling of cold-formed

steel lipped channel beams at elevated temperatures.

Keywords: Cold-formed steel structures, Lipped channel beams, Lateral-torsional buckling,

Finite element analyses, Fire design rules, Elevated temperatures

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1. Introduction

The use of cold-formed steel sections as structural members in buildings has increased in

recent times. Providing fire safety for such buildings is a vital part that should commence at

the design phase. However, there are inadequate fire design guidelines for cold-formed steel

members. Therefore there is an urgent need to investigate the behaviour of cold-formed steel

flexural members under fire conditions. Lateral-torsional buckling behaviour of cold-formed

steel beams is complicated at elevated temperatures, and a good understanding and

knowledge of this topic is vital. Although considerable research has been conducted on the

lateral-torsional buckling behaviour of hot-rolled steel members [1-10] and stainless steel

beams [11] at elevated temperatures, limited research has been reported on such behaviour of

cold-formed steel beams at elevated temperatures. Hence this research is focused on the

lateral-torsional buckling behaviour of cold-formed steel beams under simulated fire

conditions.

In this research, the section and member moment capacities of cold-formed steel lipped

channel beams at uniform elevated temperatures were investigated using an extensive

parametric study based on finite element analyses. Details of the study on the section moment

capacities at elevated temperatures are reported in Dolamune Kankanamge [12] and the

results were used in this paper in developing suitable design rules for the member capacities

of cold-formed steel beams. The applicability of relevant design rules given in Eurocode 3

Part 1.3 [13] and the new design rules developed by Dolamune Kankanamge and Mahendran

[14] based on AS/NZS 4600 [15] for ambient temperature design was investigated for fire

situations with appropriately reduced mechanical properties based on the parametric study

results for lateral-torsional buckling. The accuracy of the fire design method in Eurocode 3

Part 1.2 [16] and the proposed method of Dharma and Tan [10] for hot-rolled steel beams

was also investigated.

This paper presents the details of this research into the lateral-torsional buckling behaviour

and strength of cold-formed steel lipped channel beams at uniform elevated temperatures, its

results and comparisons with available ambient temperature and fire design rules, and the

new design rules developed in this research.

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2. Details of Finite Element Models Used in the Parametric Study

A simply supported cold-formed steel Lipped Channel Beam (LCB) under a uniform bending

moment was used in the parametric study to investigate its lateral-torsional buckling

behaviour and member capacities at ambient and elevated temperatures. A short beam model

with four-point loading was also used to investigate the section moment capacity of LCBs at

elevated temperatures and its details and results are presented in [12]. Due to the presence of

symmetric conditions in loading, support and geometry of the beams, only half the span was

modelled. Four noded S4R5 shell elements of 5 mm x 10 mm were used in the models. A

series of tensile and compressive forces was applied to the nodes at one end creating a

triangular distribution of forces across the section and thus a uniform bending moment along

the span (Figure 1). The beam was modelled with idealized simply supported boundary

conditions at the support, which allows major and minor axis rotations and warping

displacement while preventing in-plane and out-of-plane translations and twisting. It was

achieved by providing “126” boundary condition to all the nodes at the support (Figure 1).

The symmetric plane was simulated by applying boundary condition “345” to all the nodes at

the other end of the model.

The yield strength and elastic modulus of steel decrease with increasing temperatures, and

therefore appropriately reduced mechanical properties at elevated temperatures were used for

the chosen steel grades (G250 and G450) and thicknesses. The reduced yield strength and

elastic modulus at elevated temperatures were obtained by using the reduction factor

equations developed by Dolamune Kankanamge and Mahendran [17], and the measured

ambient temperature mechanical properties from tensile coupon tests. Table 1 presents the

calculated values of yield strength and elastic modulus at elevated temperatures. The stress-

strain graphs were based on the model proposed by Ranawaka and Mahendran [18] with the

recommendations made by Dolamune Kankanamge and Mahendran [17]. The strain

hardening material model was used for steels with gradual yielding type stress-strain curve

except for G250 steels at 200ºC, which have a stress-strain curve with a well defined yield

point. In the latter case, an elastic-perfect plastic material model was used.

The initial geometric imperfections and residual stresses were included in the non-linear

analyses. The corresponding lateral-torsional buckling mode obtained from the elastic

buckling analyses (Figure 2(a)) was used to input the initial geometric imperfection in the

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non-linear analysis. Beams with a negative overall geometric imperfection as shown in

Figure 2(b) yield lower moment capacities compared to positive imperfections at ambient

temperature [15,19-21]. This is also applicable to elevated temperatures and therefore a

negative geometric imperfection of L/1000 was used.

Flexural type residual stresses present in cold-formed steel beams were also included in the

non-linear analysis although they rapidly decrease with increasing temperatures. They were

calculated using an equation developed by Lee et al. [22] and the ambient temperature

flexural residual stresses. The ambient temperature flexural residual stresses were assumed to

be equal to 0.17fy along the flanges and webs and 0.08fy along the lips, based on Schafer and

Pekoz [23] and Ranawaka [24]. They were assumed to vary linearly across the thickness with

compression on the inside surface and tension on the outside surface of the section.

The finite element models were first created by using MD/PATRAN pre-processing facilities

and the analyses were then undertaken by using ABAQUS. The results were viewed by using

MD/PATRAN post-processing facilities. The developed finite element models were validated

using available numerical and experimental results as reported in [12].

Nine LCB cross-sections made of 1.5 mm and 1.9 mm thick G450 steels and 1.55 and 1.95

mm thick G250 steels were selected in the parametric study at seven temperatures ranging

from 20 to 700ºC. This selection was made so that their plate elements are compact at all the

elevated temperatures based on AS/NZS 4600 [15] calculations using the mechanical

properties at corresponding temperatures. Table 2 shows the selected LCB sections and their

dimensions. The beam span was varied from 1000 to 5000 mm, which represent the range of

beam spans failing by lateral-torsional buckling. Beams undergoing local, distortional or

lateral-distortional buckling were not considered.

3. Member Moment Capacities of LCBs Subject to Lateral-torsional Buckling at Elevated Temperatures

The developed finite element model described in Section 2 was used to conduct an extensive

series of finite element analyses (FEA) of simply supported cold-formed steel LCBs at

elevated temperatures. In total 1060 elastic buckling and nonlinear analyses were conducted

in this parametric study. The ultimate failure moments (Mu) and the elastic lateral-torsional

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buckling moments (Mo) of LCBs were obtained from FEA at varying temperatures from 20 to

700ºC. Figure 2 shows the elastic lateral-torsional buckling and ultimate failure modes of

LCBs at elevated temperatures of 300 and 600ºC.

The ultimate failure moments of LCBs were plotted in a non-dimensional format of ultimate

moment capacity (Mu/My) versus beam slenderness (My/Mo)0.5 together with the section

moment capacity results, where My is the yield moment capacity (Figure 3). Since the

selected cross-sections are compact at all the temperatures, the section moment capacity (Ms)

is higher than My. However, the use of inelastic reserve capacity of cold-formed steel beams

cannot be allowed at elevated temperatures without a detailed experimental study. Therefore

Ms was taken as equal to My at corresponding temperatures and this approach was used in

plotting the moment capacity results in a non-dimensional format. It can be seen that the

moment capacity results are scattered around the intermediate slenderness region, but

converge with increasing slenderness. The ultimate moment capacities of beams with

intermediate slenderness are reduced considerably relative to the elastic buckling moments

due to the influence of residual stresses and initial imperfections (Figure 3). With increasing

beam slenderness the ultimate capacity approaches the elastic buckling moment.

Figure 4 shows the ultimate moment capacity results at elevated temperatures for two

sections. In the case of G450 steel beams, the results for 20ºC are located above other results

while those for 700ºC provide the lower bound. In contrast, the results for 20ºC provide the

upper bound while those for 300ºC provide the lower bound for G250 steel beams. This

means that the ultimate member moment capacity results are not dependent purely on the

temperature. The reasons for the variations seen in the intermediate slenderness region are

due to other reasons discussed in the following sections.

3.1. Effect of Elevated Temperatures

As the temperature increases the yield strength and elastic modulus decrease and hence the

lateral-torsional buckling capacity also decreases as shown in Figure 5 for Grade 250 and 450

steel LCB sections. It can be seen that at 300ºC the member moment capacity of G250 steel

beam reduces considerably at lower lengths and as the length increases this strength reduction

decreases. On the other hand the member moment capacity at temperatures above 300ºC is

considerably high for Grade 450 steel beams and shows different behaviour relative to G250

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steel beams. Beyond 400ºC the member moment capacity of G450 steel beams decreases

considerably.

Figure 6 shows the relative reduction in the ultimate moment capacities at elevated

temperatures for G250 and 450 steel beams in comparison to the ambient temperature

ultimate moment capacity (Mu,T/Mu,20). At 200ºC both G250 and 450 steel beams have about

90% of their ambient temperature moment capacity. The ultimate moment capacity of G250

steel beams at 300ºC has reduced by almost 50% at lower lengths but is about 70% of the

ambient temperature moment capacity at higher lengths. In contrast to G250 steel beams at

300ºC, G450 steel beams retain 85% of the ambient temperature moment capacity at lower

lengths while their capacity is about 70% at higher lengths. The reason for this behaviour will

be discussed in the following sections. Beams of both steel grades with longer spans retain

about 45% of their ambient temperature moment capacities at 500ºC, which is considerably

high. At 700oC the moment capacities have reduced considerably and are about 18% on

average for both G250 and G450 steel beams.

3.2. Effect of Steel Grade

Figures 7 (a) and (b) show the effects of steel grade, in other words, the yield strength, elastic

modulus and the stress-strain relationship, using the results of G250 and G450 steel beams.

The results of G250 steel beams appear to be more scattered in the intermediate slenderness

range than those of G450 steel beams. The moment capacity data points for ambient

temperature are the upper bound for both G250 and G450 steel beams while those for 300ºC

in the case of G250 steel beams and 600ºC and 700ºC in the case of G450 steel beams form

the lower bound.

3.3. Effect of Non-linear Stress-strain curves

Low grade steel (G250) has a well defined yield point at ambient temperature and therefore

elastic-perfect plastic stress-strain relationship was used in FEA. It maintains this type of

stress-strain relationship up to 200ºC and becomes non-linear thereafter (Curve B in Figure

8). The stress-strain relationship of high grade steel is gradual yielding type at ambient

temperature but is very close to an elastic-perfect-plastic model (Curve A in Figure 8). This

stress-strain relationship changes with increasing temperatures and becomes non-linear

(Curve B). It is possible to assess the level of non-linearity of stress-strain relationship by

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comparing the ratio of the limit of proportionality, fp, to the 0.2% proof stress, fy. The

proportional limit of G250 steels up to 200ºC was assumed to be equal to their yield strengths

although there is a small difference. For G250 steels above 200ºC and G450 steels at all the

temperatures fp,T values were obtained by plotting the stress-strain curve using Ranawaka and

Mahendran’s [18] model and the recommendations of Dolamune Kankanamge and

Mahendran [17] with calculated values of yield strength and elastic modulus at elevated

temperatures. Figure 9 shows how the fp,T/fy,T ratio changes with increasing temperatures for

different grades of steels while Table 3 gives the values of fp,T/fy,T ratio at various elevated

temperatures.

Figure 9 shows that a different relationship exists between the fp,T/fy,T ratios of G250 and

G450 steels. The fp,T/fy,T ratio of G250 steels is constant at 1.0 up to 200ºC since the stress-

strain graph has a well defined yield point. It suddenly drops to 0.48 on average at 300ºC,

which means the material is linear up to only half of the measured yield stress. As the

temperature increases beyond 300ºC, the fp,T/fy,T ratio also increases. The lowest fp,T/fy,T ratio

occurs at 300ºC for G250 steels, which is considerably low relative to others. The fp,T/fy,T

ratio of G450 steels does not vary much as for G250 steels.

The differences in fp,T/fy,T ratio at elevated temperatures are also reflected in the ultimate

moment capacities of G250 and G450 steel beams in Figures 3 and 4. They could be seen in

the intermediate beam slenderness region. The moment capacity data points are scattered in

the case of G250 steel beams, which have a larger variation in the fp,T/fy,T ratio compared to

G450 steels. As seen in Figure 3 the results of non-dimensional moment capacities are the

lowest at 300ºC for G250 steel beams when it has the lowest fp,T/fy,T ratio. As the fp,T/fy,T ratio

increases the moment capacity data points plot higher in the intermediate slenderness range.

Therefore the effect of fp,T/fy,T ratio on the ultimate moment capacity is clearly evident.

The influence of non-linear stress-strain relationship at elevated temperatures was

investigated by conducting finite element analyses using the strain hardening material model,

which represents the true material behaviour, and the elastic-perfect plastic model with 0.2%

proof stress at 300ºC for both G250 and G450 steel beams. The ultimate moment capacities

based on these two stress-strain relationships are compared in Figures 10 (a) and (b) for G250

and G450 steel beams. The ultimate moment capacities decreased when the actual material

behaviour was used in the intermediate slenderness range in comparison to elastic-perfect-

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plastic material model for both G250 and G450 steel beams. A large reduction is seen for

Grade 250 steel beam, which has a very low fp,T/fy,T ratio whereas the difference is small for

G450 steel beam, which has a considerably high fp,T/fy,T ratio. This behaviour could be further

explained using Figure 11 in which the strain hardening and the elastic-perfect material

models are illustrated.

When the stress-strain curve is non-linear the tangent modulus is changing considerably with

the stress level (Figure 11). Beyond the elastic region the change in tangent modulus is

directly affecting the beam stiffness and therefore reduces the lateral-torsional buckling

capacity considerably. If the stress-strain relationship is highly non-linear, the use of elastic-

perfect plastic material model will lead to unsafe strength predictions. This study was

therefore conducted using strain hardening material models due to the presence of highly

nonlinear stress-strain relationship at elevated temperatures.

Recently, Dharma and Tan [10] discussed the effect of temperature related non-linear stress-

strain relationship on the lateral-torsional buckling capacity of hot-rolled steel I-beams. They

stated that the tangent modulus is very important, especially in the intermediate slenderness

range as the ultimate failure may occur beyond the limit of proportionality. They presented a

new design approach by including the non-linear stress-strain relationship in terms of fp,T/fy,T

ratio at elevated temperatures.

3.4. Effect of ky,T/kE,T Ratio

As the temperature increases the yield strength reduction rate is different from that of elastic

modulus reduction rate in relation to their ambient temperature values. Therefore the ratio of

yield strength and elastic modulus reduction factors also varies with increasing temperature.

Figure 12 shows how the ky,T/kE,T ratio varies with temperature, where ky,T (=fy,T/fy,20) and kE,T

(=ET/E20) are the yield strength and elastic modulus reduction factors at temperature T. Table

4 presents the ky,T/kE,T ratios together with ky,T and kE,T at varying temperatures. The ky,T/kE,T

ratio of G250 steels increases initially until 200ºC and thereafter reduces while it increases

until 300ºC for G450 steels and thereafter reduces until 600ºC. It then increases slightly at

700ºC.

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Since the non-dimensional member slenderness at elevated temperatures (fire conditions)

(LT,fi=√(My,T/Mo,T)) is a function of fy,T/ET, the variation of ky,T/kE,T will influence the beam

slenderness. In order to study the effect of ky,T/kE,T on LT,fi the FEA results of two G250 and

G450 steel beams with different lengths are plotted against increasing temperature in Figure

13. It is interesting to note that LT,fi for a given span that directly affects the moment

capacity does not reduce with increasing temperature. Instead LT,fi closely follows the

variation of kyT/kET ratio shown in Figure 12. For G450 steel beams the lowest LT,fi value

occurs at 600ºC when it has the lowest value of kyT/kET ratio. The highest LT,fi value for

G250 steel beams occurs at 200ºC, which has the highest ky,T/kE,T ratio.

Figures 14 (a) and (b) display the FEA results of G250 and G450 beams at various elevated

temperatures as the non-dimensionalised ultimate member moment capacity ratio (i.e.

Mu,T/My,T) versus span of the beams. The maximum ultimate member moment capacity ratio

is achieved at 600ºC when the kyT/kET ratio is the lowest while the minimum ultimate member

moment capacity ratio is achieved at 300ºC when the kyT/kET ratio is the highest for G450

steels. At longer spans, the ultimate member moment capacity ratio is increasing as the

ky,T/kET ratio is decreasing. This order of member moment capacity curves is seen, especially

for very long beams and the trend changes for beams with intermediate spans. For G250 steel

beams this behaviour can be clearly seen in Figure 14(b). The trend changes could be seen for

those with non-linear stress-strain relationships in comparison to the stress-strain

relationships with a well defined yield point at 20ºC and 200ºC. This occurs because the

increasing non-linearity of stress-strain relationship at elevated temperatures reduces the

member moment capacity as discussed in the previous section. Finally it can therefore be

concluded that for a particular span the member moment capacity achievable relative to the

section moment capacity is dependent on the kyT/kET ratio. However, this behaviour can be

altered if the non-linearity in the stress-strain relationship becomes important.

4. Comparison of Ultimate Member Moment Capacities with Predictions from the Current Design Rules

AS/NZS 4600 [15], Direct Strength Method (DSM), BS 5950 Part 5 [26] and Eurocode 3 Part

1.3 [13] are the design codes of cold-formed steel structural members at ambient temperature.

They may be used for elevated temperature design by replacing the ambient temperature

mechanical properties with those at elevated temperatures. However, the accuracy of this

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approach must be verified. AS/NZS 4600, DSM and BS5950 Part 5 gave unsafe predictions

of the member capacity of cold-formed LCBs subject to lateral-torsional buckling at ambient

temperature [14], and therefore they were not considered for elevated temperature conditions.

However, Eurocode 3 Part 1.3 [13] predictions at ambient temperature based on the buckling

curve ‘b’ of Eurocode 3 Part 1.1 [25] were overconservative in the intermediate slenderness

region while giving accurate results in the higher slenderness region. Dolamune Kankanamge

and Mahendran [14] therefore recommended buckling curve ‘a’ for the design of cold-formed

steel lipped channel beams (LCB) subject to lateral-torsional buckling at ambient

temperature. They also proposed a new ambient temperature design method with three

options within the provisions of AS/NZS 4600, and their details are given in [14]. Eurocode 3

Part 1.2 [16] is the only design code that provides suitable fire design rules for cold-formed

steel members although they were based on the studies of hot-rolled steel members [1-4].

In this section, the accuracy of the design methods given in Eurocode 3 Part 1.3/Part 1.1 and

Part 1.2, and the new ambient temperature design equations of Dolamune Kankanamge and

Mahendran [14] was assessed by comparing their predicted member moment capacities with

the FEA results obtained at different elevated temperatures. Dharma and Tan [10] proposed a

design method for hot-rolled beams under fire conditions and its accuracy was also assessed

for cold-formed steel beams. When the ambient temperature design rules were used to predict

the ultimate member moment capacities at elevated temperature conditions, the reduced

mechanical properties in Table 1 were used in all the calculations.

4.1. Eurocode 3 Part 1.3 / Eurocode 3 Part 1.1

The ambient temperature design code for cold-formed steel members, Eurocode 3 Part 1.3

[13], refers to the design equations given in the design code for hot-rolled steel members,

Eurocode 3 Part 1.1 [25], to determine the lateral-torsional buckling capacity (Mb,Rd) using

buckling curve ‘b’. The relevant design equations are given next.

yfyWM LTRdb , (1)

where

5.0221

LTLTLTLT

2_

2.0_

15.0 LTLTLTLT (3)

(2)

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5.0_ oMyfyWLT (4)

Wy is the appropriate section modulus of the compression flange depending on the class of

cross section (plastic, elastic or effective section modulus), fy is the yield stress of steel, Mo is

the elastic lateral-torsional buckling moment and the imperfection factor LT for buckling

curve ‘b’ is 0.34. The product Wyfy is the section moment capacity (Ms), which was used in

plotting the non-dimensional moment capacity curves. Since the LCB sections considered

here included compact plate elements, Eurocode 3 Part 1.3 design rules give section moment

capacities that are higher than their first yield moments. In all the design methods used in

Section 4, the standard lateral-torsional buckling equation was used to calculate Mo at

ambient and elevated temperatures using the appropriately reduced mechanical properties in

Table 1.

Figures 15 (a) and (b) compare the FEA results of G250 and G450 cold-formed steel beams

at elevated temperatures with the moment capacity curves obtained based on the four

buckling curves in Eurocode 3 Part 1.1 [25] (Table 5). It can be seen that the use of single

buckling curve (‘a’ or ‘b’) is inaccurate for some temperatures for G450 steel beams. For

G250 steel beams the moment capacity data points are scattered in the intermediate

slenderness range and thus the use of any of these curves will give both unsafe and

overconservative predictions at some temperatures. For example, the data points for 300ºC

and 400ºC are plotted below even the buckling curve ‘b’, i.e. the predictions of Eurocode 3

Part 1.3 [13] using buckling curve ‘b’ are unsafe. Therefore suitable buckling curves were

proposed for different temperature ranges and G250 and G450 steel beams as shown in

Tables 6 (a) and (b).

Figures 16 (a) to (c) show the comparison of ultimate member moment capacities of G450

steel beams from FEA with the proposed buckling curves in Table 6 for the three different

temperature ranges while Figures 17 (a) to (c) show the comparison for G250 steel beams.

These figures show that the chosen buckling curves give reasonably accurate predictions, but

further improvements are needed for some temperatures. Table 7 shows the mean and COV

of the FEA to predicted member moment capacities using the new proposal based on

different buckling curves. On average the proposal has a mean of 1.046 with a COV of 0.103.

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It also includes the corresponding capacity reduction factors calculated using the North

American Specification [27] procedure to provide the recommended target reliability index of

2.5 for cold-formed steel beams. This procedure is based on a statistical model to account for

the variations in material, fabrication and loading effects. In these calculations suitable values

of mean and COV of the material, fabrication and loading factors were used as recommended

in [27]. The overall capacity reduction factor is 0.915.

This proposal does not include the effect of the non-linear stress-strain characteristics at

elevated temperatures that significantly influences the ultimate moment capacities of cold-

formed steel beams. Therefore a new design method is required that includes this effect. It

should allow the same design method to be used for beams made of any type of steel.

4.2. New Ambient Temperature Design Method Proposed by Dolamune Kankanamge and Mahendran [14]

In this method, the nominal member moment capacity (Mb) of laterally unbraced beams

subjected to lateral buckling is given by,

f

ccb Z

MZM (5)

where Zc is the effective section modulus calculated at a stress level (Mc/Zf) and Zf is the full

unreduced section modulus. Since the selected LCBs are compact, Zc is equal to Zf.

The critical moment (Mc) in Equation 5 is calculated as follows,

For 6.0b yc MM (6a) For 6.0b ybbc MM 24 05.29.0 (6b) where

5.0

o

yb M

M

The ultimate moment capacities obtained from FEA at elevated temperatures are then

compared in Figures 18 (a) and (b) with the new design method developed for ambient

temperature design [14]. This comparison shows that the new design method is accurate for

some temperatures, but gives unsafe predictions at 300ºC, 400ºC and 600ºC for G250 steel

beams and 600ºC and 700ºC for G450 steel beams. The scattered nature of the results in the

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intermediate slenderness region does not allow the use of a single beam design curve for all

the temperatures. Therefore separate beam design curves are required to accurately represent

the behaviour at elevated temperatures.

4.3. Eurocode 3 Part 1.2

The design lateral-torsional buckling resistance moment of a laterally unrestrained beam at

elevated temperatures (T) is given by,

yTyyfLTtRdfib fkWiM ,,,, (7)

where Wy and fy are as defined in Eurocode Part 1.3 (ie. at ambient temperature), and kY,T has

also been defined previously.

5.02

,2

,,1, TLTTLTTLTfiLT

(8)

with 212

1,,, TLTTLTTLT (9)

The imperfection factor is given by,

yf23565.0 (10)

The non-dimensional slenderness TLT , is given by,

5.0,,, TETyLTTLT kk (11) 5.0oyyLT MfW is the beam slenderness at ambient temperature as in Eurocode 3 Part 1.3.

The lateral-torsional buckling capacity of beams at elevated temperatures is affected by

residual stresses, initial crookedness and twist, material reduction factors and nonlinear

stress-strain relationship. Most of these factors are different in cold-formed steel sections in

comparison to hot-rolled steel sections and therefore considerable differences are possible

between the behaviour of these two types of steel beams. Therefore the applicability of hot-

rolled steel based fire design method must be verified for cold-formed steel members.

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The section moment capacity Ms used in the calculations of this method is Wyky,Tfy at varying

temperatures. Figures 19 (a) and (b) show the comparison of FEA results obtained for

different elevated temperatures with the beam design curve obtained from the fire design

code, Eurocode 3 Part 1.2. Most of the results are above the design curves for G250 and

G450 steel beams. For some temperatures the capacity predictions are over-conservative.

This strongly raises the need for multiple design curves for steel beams subject to lateral-

torsional buckling as a function of elevated temperatures. Eurocode 3 Part 1.2 [16] predicts

the moment capacities reasonably well for beams with higher slenderness (i.e.

(Ms/Mo)0.5≥2.0) for both G250 and G450 steel beams. Therefore it can be concluded that for

intermediate slenderness values, (Ms/Mo)0.5

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“Rankine Approach”. Hence this study considers the Alternative Approach, and the ultimate

moment capacity (Mb,fi) at elevated temperatures is given next.

xfibfib SfM ,, (12) where,

xS - the plastic section modulus about the major axis.

TEyTyTLTTLT

TEyTy

fibffk

ffkf

,,,

,, 2

(13)

in which,

21,,

,

TETLTyTy

TLT

ffk

and (14)

2,

2

LT

TETE

Ekf

(15)

E

sx

yLT M

MfE2 (16)

In Equation 16, sxM is the in-plane bending moment capacity. EM is the elastic buckling

moment capacity considering major axis (x) curvature given by Kirby and Nethercot [28] as,

xy

oE EIEI

MM

1

(17)

The initial curvature and twist term TLT , in Equation 14 at elevated temperature is given by

04.0007.0,

,5.12

,

Ty

TE

E

cx

yTLT k

kMM

fE (18)

They used a non-linearity factor to account for the non-linear stress-strain relationship at elevated temperatures as discussed in Section 3.3.

Ty

Tp

ff

,

, (19)

The FEA results are compared in Figure 20 for one LCB section for several temperatures

using Dharma and Tan’s alternative approach. It can be seen that the capacity predictions

using this method are unsafe at some temperatures, especially at 600ºC for G450 steel beams

and 300ºC for G250 steel beams. Dharma and Tan [10] developed their rules based on doubly

symmetric hot-rolled steel beams whose stress-strain relationships at elevated temperatures

are very different to those of cold-formed steel beams. This is the possible reason for the

disagreement in the results.

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5. New Proposal

Previous sections showed the need for a new design method for the fire design of cold-

formed steel beams subject to lateral-torsional buckling. The current design method given in

the fire design code, Eurocode 3 Part 1.2 [16], was found to be over-conservative while it was

found that separate buckling curves are needed for different elevated temperature ranges

when using Eurocode 3 Part 1.3. The FEA ultimate moment capacities were found to be

dependent on the level of non-linearity in the stress-strain curve, particularly in the

intermediate slenderness region. It was found that the effect of non-linearity can be accounted

for by using the fp,T/fy,T factor and therefore new design equations were proposed by

including this factor in the ambient temperature design equations proposed by Dolamune

Kankanamge and Mahendran [14].

The nominal member moment capacity ( TbM , ) of the laterally unbraced cold-formed steel

beams subject to lateral-torsional buckling at elevated temperatures (T) is given by,

Tf

TcTcTb Z

MZM

,

,,, (20)

where Zc,T and Zf,T are the section properties at elevated temperatures and Mc,T is given by,

For 6.0, Tb TyTc MM ,, (21a)

For 6.0, Tb TyMTbf

fTbf

fTcM

Ty

Tp

Ty

Tp

,2,05.2

4,9.0,

55.0

,

,

4.0

,

,

(21b)

where,

Ty

Tp

ff

,

, - the ratio of proportional limit and yield strength at temperature T.

5.0

,

,,

To

TyTb M

M (22)

TyM , in Equations 21 (b) and 22 is the yield moment capacity at temperature T given by,

a b

17

TyTfTy fZM ,,, (23) Elastic lateral-torsional buckling moment at temperature T is given by,

2

2

2

2

, LIEJG

LIE

M wTTyT

To (24)

TE and TG are the elastic modulus and the shear modulus at temperature T, Iy, Iw and J are the

section properties, and L is the span. As defined in Section 3.4. elastic modulus and yield

strength at temperature T are given by 20, EkE TET and 20,,, yTyTy fkf .

In this method developed within the provisions of AS/NZS 4600 [15], the section moment

capacity of the selected LCB sections is limited to their yield moment capacity although they

are compact (see Eq.21a). The ultimate moment capacities at elevated temperatures from

FEA were compared with the predicted moment capacities with appropriately reduced

mechanical properties for each temperature using the proposed new design equations. The

results are plotted in the non-dimensionalised format for different temperatures in Figures 21

and 22 for Grade 250 and Grade 450 steel beams, respectively. Suitability of the proposed

design method was investigated for each temperature and it could be seen that the new

proposal with reduced mechanical properties is able to predict the ultimate member moment

capacities accurately.

Table 8 presents the mean and COV of the ratio of FEA to predicted moment capacities using

the developed design equations for varying elevated temperatures. The overall mean and

COV are 0.983 and 0.068 while the associated capacity reduction factor is 0.886. Therefore it

is recommended that the proposed design method can be used for cold-formed steel lipped

channel beams subject to lateral-torsional buckling at elevated temperatures.

The new design method has many advantages in comparison to other design methods,

especially the fire design method in Eurocode 3 Part 1.2. The new proposed equation is a

modification of ambient temperature design method and is the same if used for ambient

temperature design because at ambient temperature the material non-linearity factor is almost

1.0. Another advantage is that the proposed equation can be modified by changing the values

of ‘a’ and ‘b’ as shown in Equation 21b to have different buckling curves and provides a

unique way of developing design methods for other types of cross-sections. This design

18

method provides a much simpler way of calculating the lateral-torsional buckling capacities

at elevated temperatures than other methods.

6. Conclusions

This paper has described a detailed parametric study on the behaviour of cold-formed steel

lipped channel beams (LCB) subject to lateral-torsional buckling at elevated temperatures.

The parametric study was undertaken for nine LCB sections using validated finite element

models. It was shown that the lateral-torsional buckling capacity is strongly influenced by the

level of non-linearity of the stress-strain curves at elevated temperatures and that the level of

non-linearity can be represented by the ratio of the limit of proportionality to the yield

strength of steel.

The ultimate member moment capacities from the parametric study were compared with

those predicted by the ambient temperature design code, Eurocode 3 Part 1.3, the new design

method developed by Dolamune Kankanamge and Mahendran [14], the fire design code,

Eurocode 3 Part 1.2 and Dharma and Tan’s [10] design method. These comparisons showed

that using a single buckling curve is inadequate to obtain the ultimate moment capacities of

cold-formed steel beams at varying elevated temperatures. Hence Eurocode 3 Part 1.3 design

method with buckling curve ‘b’ and the method proposed by Dolamune Kankanamge and

Mahendran [14] are unsafe or overconservative for some temperatures, especially in the

intermediate slenderness region. Therefore other buckling curves in Eurcode 3 Part 1.1 were

proposed for different temperature ranges for the fire design of cold-formed steel LCBs.

Eurocode 3 Part 1.2 predictions were found to be overconservative for all the temperatures

except for beams with very high slenderness. The design method proposed by Dharma and

Tan [10] provided reasonably accurate capacity predictions for some temperatures. Finally, a

new design method was developed for the design of cold-formed steel LCBs subject to

lateral-torsional buckling at elevated temperatures. This design proposal enables safe

structural designs of cold-formed steel beams in fire situations.

Acknowledgements

19

The authors would like to thank Australian Research Council for their financial support and

the Queensland University of Technology for providing the necessary research facilities and

support to conduct this research project.

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